I currently ride a hybrid that came with 700x32c tires. Can I swap out a narrower tire without changing the rims, or do rims only fit a specific size of tire? How small can I go before I have to buy new rims? I hear having narrower tires makes you go quite a bit faster. Does it really help that much?
Most 700c rims will work with essentially any size of tire. Yes, smaller tires generally reduce rolling resistance a little, but unless you're road racing, it's rarely worth going below about 700x23c, IMO. Also note that just being smaller doesn't necessarily mean lower rolling resistance -- one brand of 700x19c might have higher rolling resistance than another brand of 700x20c, for example.
The big change (if any) happens when you switch to slick tires (and most 700x32c tires have fairly substantial tread if not outright knobs).
For smooth tires, rolling resistance is a relatively small factor in terms of your overall performance. The optimal tire depends on your riding conditions and personal riding style. Narrower does not automatically equate to faster - there are both terrible narrow tires and excellent wide tires available today. If a tire otherwise possesses the characteristics you desire - such as grip, flat resistance, cost, handling - you'll probably be just fine riding it from a pure performance perspective.
It's also categorically false that maximizing the pressure in your tires will make you faster. There is a "sweet spot" in inflation where the tire is stiff enough to roll smoothly but still able to smooth out road vibrations that lead to fatigue. Bicycle Quarterly tested and published a handy inflation cheat sheet that may help you dial in your optimal riding pressure.
As far as rim compatibility, the width of you rim will affect the shape of the tire; if you plan on running any tire narrower than 28mm you should measure your rim and consult this handy chart. It's overly-cautious but a useful rule of thumb for the new mechanic.